I suspect your professor was referring to 'qualitative' vs 'quantitative' approaches when referring to soft vs hard analysis. The baire category theorem is int

I suspect your professor was referring to 'qualitative' vs 'quantitative' approaches when referring to soft vs hard analysis. The baire category theorem is intimately related to these to approaches, and leads to three theorems (1) the uniform boundedness principle, (2) the open mapping theorem, and (3) the closed graph theorem, that establish equivalences between the qualitative and quantitative theories. According to Tao's textbook, in practice these are not used much directly, instead one starts with quantitative bounds, then derives the qualitative corollaries. The baire category theorem tells us why those there theorems are true, and thus tells us why we can take that step from quantitative to qualitative. I think this is what your professor meant.